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PureMathematicsnØêÆ,2019,9(9),1102-1107
PublishedOnlineNovemb er2019inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2019.99135
SolutionsofaClassofEulerFunction
Equations
KeliPu
DepartmentofMathematicsandCom puterScience,ABaTeachersCollege,WenchuanSichuan
Received:Nov.2
nd
,2019;accepted:Nov.21
st
,2019;published:Nov.28
th
,2019
Abstract
Letnbeapositiveinteger,ϕ(n)isEulerfunction,thevalueisequaltothesequence
0,1,2,...,n−1whichareprimeton.Infact,discussingthesolutionsofEulerfunction
equationisameaningfulwork,moreover,thepropertiesofthefunctionarevery
importanttodiscussthesolution.Inthispaper,usingthepropertiesoftheEuler
function,wediscussthenecessityofintegersolutionoftheEulerfunctionequation
ϕ(mn)=aϕ(m)+bϕ(n)+c,andthengivesallsolutionsifa=5,b=6,C=16.
Keywords
EulerFunction,PropertiesofEulerFunction,IntegerSolution
˜aî.¼ê•§)
ÆÆÆŒŒŒsss
Ct“‰Æ§êƆOŽÅ‰ÆÆ§oAfA
ÂvFϵ2019c112F¶¹^Fϵ2019c1121F¶uÙFϵ2019c1128F
©ÙÚ^:ÆŒs.˜aî.¼ê•§)[J].nØêÆ,2019,9(9):1102-1107.
DOI:10.12677/pm.2019.99135
ÆŒs
Á‡
n´ê,ϕ(n)´Í¶î.¼ê§§ŠuS0,1,2...n−1¥†npƒê‡
ê"éu9î.¼ê•§)?Ø´˜‡Lk¿Â‘K§î.¼ê5Ÿ3?Øî
.¼ê•§)¥–'-‡"©|^î.¼ê5Ÿƒ'(ا?Ø˜aî.¼ê•
§ϕ(mn)=aϕ(m)+bϕ(n)+c•3ê)7‡^‡§¿‰Ña=5,b=6,c=16ž§Tî
.¼ê•§Ü)"
'…c
î.¼ê,î.¼ê5Ÿ,ê)
Copyright
c
2019byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
n´ê§ϕ(n)´Í¶î.¼ê§§ŠuS0,1,2...n−1¥†npƒê‡ê"
'uEuler¼êϕ(n)•§´êØ¥š~-‡Úk¿Â‘K§ék'uϕ(n)5ŸÚϕ(n)k'
ؽ•§ïħNõÆö?1&?(Œëw©z[1–13])"Ù¥©z[1,2]éu•§ϕ(x)=m
)?1?ا©z[3]‰Ñm=2p,2p
n
,2pqž•§ϕ(x)=m)(Ù¥p,q•ƒê§n•
ê)"3©z[4–7]¥K©O?Ø•§ϕ(mn)=k(ϕ(m)+ϕ(n))Ù¥k∈Z§3k
ØÓŠž)"é/Xϕ(mn)=aϕ(m)+bϕ(n)+cî.¼êš‚5•§§©z[13]‰Ñ
a=7,b=8,c=16ž•§Ü)"©?Ø•§ϕ(mn)=aϕ(m)+bϕ(n)+c(Ù
¥a,b,c∈Z)ê)§¿‰Ñϕ(mn)=5ϕ(m)+6ϕ(n)+16Üê)"
2.î.¼ê5Ÿƒ'(J
Ún1[14]m,n•?¿ê§em|n§Kϕ(m)|ϕ(n)"
Ún2[14]é?¿êm,n§egcd(m,n)=d§Kϕ(mn)=
dϕ(m)ϕ(n)
ϕ(d)
"
Ún3[14]n≥1ž§ϕ(n)≤n§n≥3ž§ϕ(n)7•óê"
íØ4é?¿n‡êx
1
,x
2
,...,x
n
§k
ϕ(x
1
x
2
...x
n
)≥ϕ(x
1
)ϕ(x
2
)...ϕ(x
n
)
DOI:10.12677/pm.2019.991351103nØêÆ
ÆŒs
y²dÚn29Ún3á"
Ún5[15]é?¿ên,p´ƒê§K
ϕ(np)=
(
(p−1)ϕ(n),(n,p)=1,
pϕ(n),(n,p)=p.
3.̇(J9y²
Ún6[3]p•ƒê§ϕ(x)=2p)•
(1)p=2ž§x=5,8,10,12
(2)p=3ž§x=7,9,14,18
(3)p≥5ž§g=2p+1•ƒê§ϕ(x)=2pkü‡)x=g,2g¶g=ϕ(2p+1)Ø•ƒêž§
Kϕ(x)=2pÃê)"
Ún7[3]eϕ(x)=2,Kx=3,4,6"
ϕ(x)=2
2
§Kx=5,8,10,12.
ϕ(x)=2
3
§Kx=15,16,20,24,30.
ϕ(x)=2
4
§Kx=17,32,34,40,48,60.
ϕ(x)=2
5
§Kx=51,64,68,80,96,102,120.
½n6î.¼ê•§ϕ(mn)=aϕ(m)+bϕ(n)+c(Ù¥a,b,c∈Z)§e•3ê)(m,n)§
Kϕ(gcd(m,n))|c"
y²Ø”gcd(m,n)=d§Kd|m,d|n"dÚn1Œϕ(d)|ϕ(m)…ϕ(d)|ϕ(n)"-ϕ(m)=
m
1
ϕ(d),
ϕ(n)=n
1
ϕ(d),Ù¥m
1
,n
1
∈Z
+
§2dÚn2ϕ(d)(dm
1
n
1
−am
1
−bn
1
)=c§=ϕ(d)|c§y"
½n7î.¼ê•§ϕ(mn)=5ϕ(m)+6ϕ(n)+16ê)21|§©O•
(m,n)=(53,7),(53,9),(53,14),(53,18),(106,7),(106,9),
(15,29),(16,29),(20,29),(24,29),(30,29),(15,58),(8,38),
(8,54),(10,38),(10,54),(12,38),(75,12),(12,18),(20,10),(30,10).
y²-gcd(m,n)=d§dÚn2Œϕ(d)(dm
1
n
1
−5m
1
−6n
1
)=16§2d½n6Œϕ(d)|16§
Kϕ(d)=1,2,4,8,16"e¡©œ¹?ص
I.ϕ(d)=1§Kd=1½2"
d=1ž§Km
1
n
1
−5m
1
−6n
1
=16§=(m
1
−6)(n
1
−5)=46ddŒ(m
1
,n
1
)=
(7,51),(52,6),(8,28),(29,7)§(m
1
,n
1
)=(7,51),(29,7)ž§duϕ(d)=1§¤±ϕ(m)ϕ(n)þ
•Ûê§ù†Ún3gñ§¤±(m
1
,n
1
)=(52,6),(8,28)"(m
1
,n
1
)=(52,6)ž§džϕ(m)=52§
DOI:10.12677/pm.2019.991351104nØêÆ
ÆŒs
KOŽŒm=53,106§ϕ(n)=6§Kn=7,9,14,18,¤±(m,n)=(53,7),(53,9)(53,14),(53,18),
(106,7),(106,9)"(m
1
,n
1
)=(8,28)ž§ϕ(m)=8,Km=15,16,20,24,30"ϕ(n)=28§Kn=
29,58"¤±
(m,n)=(15,29),(16,29)(20,29),(24,29),(30,29),(15,58).
d=2ž§K2m
1
n
1
−5m
1
−6n
1
=16§=(m
1
−3)(2n
1
−5)=31,ddŒ(m
1
,n
1
)=
(4,18),(34,3)"(34,3)Ø÷v^‡§"=(m
1
,n
1
)=(4,18)§Kϕ(m)=4§Km=5,8,10,12"
ϕ(n)=18§Kn=19,27,38,54"¤±
(m,n)=(8,38),(8,54),(10,38),(10,54),(12,38).
II.ϕ(d)=2§Kd=3,4,6"
d=3ž§K3m
1
n
1
−5m
1
−6n
1
=8§=(3m
1
−6)(3n
1
−5)=54"K(m
1
,n
1
)=(20,2)§d
žϕ(m)=40§Km=41,55,75,82,88,100,110,132,150,ϕ(n)=4§Kn=5,8,10,12"¤±
(m,n)=(75,12)
d=4ž§K4m
1
n
1
−5m
1
−6n
1
=8§=(2m
1
−3)(4n
1
−5)=31§K(m
1
,n
1
)=(2,9)§d
žϕ(m)=4§Km=5,8,10,12”ϕ(n)=18,Kn=19,27,38,54"Ïgcd(m,n)=4§¤±•§Ã
ê)"
d=6ž§K6m
1
n
1
−5m
1
−6n
1
=8§=(2m
1
−2)(6n
1
−5)=26§K(m
1
,n
1
)=(14,1),(2,3)"
(m
1
,n
1
)=(14,1)§džϕ(m)=28§Km=29,58.ϕ(n)=2§Kn=3,4,6§gcd(m,n)=6§
dž•§Ãê)"(m
1
,n
1
)=(2,3)§džϕ(m)=4§Km=5,8,10,12.ϕ(n)=6§Kn=
7,9,14,18"¤±
(m,n)=(12,18).
III.ϕ(d)=4§Kd=5,8,10,12"
d=5ž§5m
1
n
1
−5m
1
−6n
1
=4§=(5m
1
−6)(n
1
−1)=10§džØ•3m
1
,n
1
∈Z
+
¦
ª¤á§•§Ãê)"
d=8ž§8m
1
n
1
−5m
1
−6n
1
=4§=(4m
1
−3)(8n
1
−5)=31§džØ•3m
1
,n
1
∈Z
+
¦
ª¤á§•§Ãê)"
d=10ž§10m
1
n
1
−5m
1
−6n
1
=4§=(5m
1
−3)(2n
1
−1)=7§K(m
1
,n
1
)=(2,1)§d
žϕ(m)=8§Km=15,16,20,24,30.ϕ(n)=4§Kn=5,8,10,12"¤±
(m,n)=(20,10),(30,10).
d=12ž§12m
1
n
1
−5m
1
−6n
1
=4§=(2m
1
−1)(12n
1
−5)=13§džØ•3m
1
,n
1
∈Z
+
¦
ª¤á§•§Ãê)"
DOI:10.12677/pm.2019.991351105nØêÆ
ÆŒs
IV.ϕ(d)=8§Kd=15,16,20,24,30"
d=15ž§15m
1
n
1
−5m
1
−6n
1
=2§=(5m
1
−2)(3n
1
−1)=4.
d=16ž§16m
1
n
1
−5m
1
−6n
1
=2§=(8m
1
−3)(16n
1
−5)=31.
d=20ž§20m
1
n
1
−5m
1
−6n
1
=2§=(10m
1
−3)(4n
1
−1)=7.
d=24ž§24m
1
n
1
−5m
1
−6n
1
=2§=(24m
1
−5)(4n
1
−1)=13.
d=30ž§30m
1
n
1
−5m
1
−6n
1
=2§=(5m
1
−1)(6n
1
−1)=3.
dž§éþã5«œ¹þØ•3m
1
,n
1
¦ª¤á§•§Ãê)"
V.ϕ(d)=16§Kd=17,32,34,40,48,60"ÓþOŽŒ§þØ•3m
1
,n
1
¦ªdm
1
n
1
−
5m
1
−6n
1
=1¤á§•§Ãê)"nþ½ny"
Ä7‘8
I[g,‰ÆÄ7(11861001)§o AŽA^Ä:ïÄ‘8(2018JY0458)§oAŽp‰ïM #
ìèïOy(18TD0047)§Ct“‰Æ?‘8(ASB18-02)"
ë•©z
[1]Schinzel,A.(1956)Surl’Equationϕ(x)=m.ElementederMathematik,11,75-78.
[2]Ford,K.(1999)TheNumberofSolutionsofϕ(x)=m.AnnalsofMathematics,150,283-311.
https://doi.org/10.2307/121103
[3]ñlÇ.'uEuler¼ê•§ϕ(x)=m)[J].-Ÿó’+nÆÆ,1998,12(5):91-94.
[4]Üo.k'Euler¼êϕ(n)•§ê)[J].êÆ¢‚†@£,2014,44(20):302-305.
[5]N_,M….'uî.•§ϕ(ab)=2k(ϕ(a)+ϕ(b))ê)[J].ÜH“‰ŒÆÆ(g,‰
Ƈ),2016,41(4):6-9.
[6]+Sr,Üo.†Euler¼êϕ(n)k'ü‡•§[J].êÆ¢‚†@£,2016,46(9):221-225.
[7]°•,pw,…Ê.k'Euler¼êϕ(n)•§Œ)5¯K[J].ô܉Æ,2016,34(1):
15-16.
[8]•R.˜a•¹Smarandache¼êÚEuler¼ê•§[J].ÜHŒÆÆ(g,‰Æ‡),2012,
34(2):70-73.
[9]4R,œ+.'u•§
P
S(d)=ϕ(n)Œ)5.ÜHŒÆÆ(g,‰Æ‡),2013,35(6):54-58.
[10]•I¦.˜‡•¹Euler¼ê•§[J].X{êƆA^êÆ,2007,23(4):439-445.
[11]Üo,Úk•.˜‡•¹Euler¼ê•§ê)[J].u¥“‰ŒÆÆ(g,‰Æ‡),2015,
49(4):497-501.
[12]Sun,C.F.andCheng,Z.(2010)SomeKindofEquationsInvolvingEulerFunctionϕ(n).
JournalofMathematicalStudy,43,364-369.
DOI:10.12677/pm.2019.991351106nØêÆ
ÆŒs
[13]gŸ##Ü,Üo,=÷Œ.˜‡k'Euler¼êϕ(n)š‚5•§)[J].ÄÑ“‰ŒÆ
Æ(g,‰Æ‡),2018,39(2):4-7.
[14]Rosen,K.H.(2005)ElementaryTheoryandItsApplications.5thEdition,PearsonEducation
Inc.,AddisonWesley,UpperSaddleRiver,NJ.
[15]š}•,§œ.'u•§ϕ(xyz)=2(ϕ(x)+ϕ(y)+ϕ(z))[J].êÆ¢‚†@£,2012,42(23):
267-271.
DOI:10.12677/pm.2019.991351107nØêÆ

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